Jacobson radical

This article defines an ideal-defining function, viz a rule that inputs a commutative unital ring and outputs an ideal of that ring

Definition for commutative rings

Symbol-free definition

The Jacobson radical of a commutative unital ring is the intersection of all its maximal ideals.

Definition for noncommutative rings

The noncommutative case was considered by Jacobson while proving the famous Jacobson density theorem (Fill this in later).

Particular cases

Trivial ideal

A commutative unital ring whose Jacobson radical is trivial is termed a semisimple ring.